These exercises, activities and learning activities are designed for students to use independently or in small groups to practise number properties. Some involve investigation and may become longer and more involved tasks with subsequent recording/reporting. Typically an exercise is a 10 to 15 minute activity. The exercises can also be used as teaching activities with similar supporting work for students.
Multiplication and Division, AM (Stage 8)
100s chart and flipblock 100s board
Practice exercises with answers (PDF or Word)
counters
calculator
Prior Knowledge
Background
These exercises are powerful illustrations of the connectivity of mathematics. The square of a number is established and then the odd numbers are modelled and linked to the squares. The triangular numbers are modelled and the squares are linked to these. All the while patterns are being sorted and larger numbers with repetition of the smaller squares reinforced. The link of odds to squares and squares to triangular and therefore odds to triangular is discovered. This is a form of the equivalence relationship or the Zeroth Law of Thermodynamics (If A= B and B=C then A=C). The exercises then develop into using knowledge of squares. If you know 12x12 then what do you know about 13x13? While it could develop to the generalisation of (x-a)(x-b) this is not done.
Comments on the Exercises
Asks students to find the square of numbers.In this exercise the notion of square is established with the common single digit squares being practiced. Locating these on a 100s board should evoke the odd number pattern between the squares. It is important that different visualisations of these numbers are given. The reverse notion of square root is not mentioned but could be used as an advanced organiser for exercise 12. Revise the way an equals works both ways and what it means. It does not mean “evaluate this”.
Asks students to explore the associative property.
The way 20 x 20 becomes 2x10x2x10 = 2x2x10x10 is very important. This needs to be explored and established as knowledge. There is a lot of language surrounding the use of square.
The “quad” on quadrillon means 4 groups of 000 before the million so 1 quadrillion is 1,000,000,000,000,000,000 and is the same as 1000 trillion (tri meaning 3) and 1000,000 billion (bi meaning 2). A googol is from Charlie Brown cartoon series in which Schrodinger, in response to a question from Lucy, estimates the chances that he is in love with her as 1 in a googol or extremely slim. A googol is 10 to the power of 100 or 10100. It is fun to write this number out. What is a googolplex?
Exercise 3
Asks students to explore the distributive property.
The array model is extremely important as a way to solve problems. It uses the area notion and is difficult for additive students to comprehend. It is this notion that is a barrier for these students. Use it with lots of explanation. Repeat the use of the word “lots” to establish it is groups that are being used. A challenge is for students to invent their own games and so develop responsibility for their own learning.
Repeat! Additive students will find areas and squares very challenging.
Asks students to explore the relationship between odd numbers and square numbers. This property of odd numbers is a very useful and commonly occuring pattern. It is beautifully modelled as a L and joined as a square when summed from 1. Summing any consecutive odd numbers leads to the difference of two squares quite easily but should be reserved for the better student or later when it can be explored properly and fully. Other models of odd numbers using side by side shapes can be useful as well to illustrate even and oddness.
Asks students to explore patterns with odd numbers. The odd number set can be ordered in many ways. Two of these are shown and are very useful and very interesting respectively. The squares come from summing the odds. The cubes and squares come from ordering them in a counting sort of way.
Asks students to explore the pattern between square numbers. The powerful question “if you know that 12x12=144, what do you know about 13x13” promotes investigation and has several obvious answers. The answer can be additive or multiplicative or a combination of the two. All options should be discovered and made available to students. Using this idea is import as an example of what can be done. This carves a problem solving pathway which will prove useful in later mathematics. The use of the word “lots” and how they are grouped is interesting and illustrates the distributive law first by the common term and then commutatively by the common group.
Asks students to explore the way triangular numbers combine and make the squares. It is very important to explore and know how to sum the triangular numbers. Use of colours highlights the count. This can be done with all students. What they will see is limited by their numeracy stage. A counter will see just that. An adder will see patterns and may be able to make groups of equal size. A grouper will see the area model and may be able to generalise the solution. The consecutive idea of two T numbers Tn-1 and Tn making a square of nxn is only one of many such patters and others can be explored. Knowing the algebra for these sums can lead to iinteresting proof by algebraic manipulation. Eg Tn = n(n+1)/2, show Tn-1 + Tn = nxn. A curious extension is asking “Are any triangular numbers also square numbers? The answer to this is left for the curious to consider.
Asks students to explore patterns in square numbers. It is the reverse of the previous exercise 6 to point out another way to look at a problem. It is applied in some useful ways.
Asks students to solve square problems using other square number facts. It is an extension to a greater difference up (and down) from a know square number. It may be useful but is perhaps more useful as foundation to the general expansion of (x+a)(x+b). There are different ways to look at the problem depending on strategy stage. Students could be encouraged to invent their own enormous square number problems.
Asks students to explore a curious trick with square numbers.
This exercise is an introduction to the complexities of the square root. It is important that the “odd” and even” number of zeros is introduced. When finding a square root it is useful to pair up the digits and notice that the answer is often the size of the number of pairs. A 6-digit number has 3 pairs and the “root” will be in the thousands.
Exercise 13
The main purpose of doing this exercise is to gain a notion of what n+1, 2n etc means. Encourage dialogue that it all depends on the value of n. It is more important that 2n means double than where it is on the number line. Invite student problems and extend to powers and decimals.